In the last years,
new variants of the minimum cycle basis (MCB) problem
and new classes of cycle basis have been introduced,
as motivated by several applications from disparate
areas of scientific and technological inquiries.
At present, the complexity status of the MCB problem
has been settled only for undirected, directed,
and strictly fundamental cycle basis.

In this paper, we offer an unitary classification
accommodating these 3 classes and further including
the following 4 relevant classes: 2-bases (or planar bases),
weakly fundamental cycle bases, totally unimodular cycle bases,
and integral cycle bases. The classification is complete in that,
for each ordered pair (A,B) of classes considered,
we either prove that A⊆B holds for every graph
or provide a counterexample graph for which A⊄B.
The seven notions of cycle bases are distinct
(either A⊄B or B⊄A
is exhibited for each pair (A,B).

All counterexamples proposed have been designed
to be ultimately effective in separating the various algorithmic
variants of the MCB problem naturally associated to each one
of these seven classes. We even provide a linear time
algorithm for computing a minimum 2-basis of a graph.
Finally, notice that the resolution of the complexity
status of some of the remaining three classes would
have an immediate impact on practical applications,
as for instance in periodic railway timetabling, only
integral cycle bases are of direct use.